Pattern Formation and Synchronism in an Allelopathic Plankton Model with Delay in a Network

Abstract

A network is introduced to describe the spatiotemporal dynamics of two-species competitive and allelopathic plankton models, where the network structure represents the movement directions between every two patches. Time delay is also incorporated to describe the time required to produce stimulatory effect of one species on the growth of the other species. The model is described by a system of discrete-space and continuous-time equations with time delay in a network. Using the time delay as a bifurcation parameter, it is shown that a Hopf bifurcation occurs in the system. The stability of the Hopf bifurcation is also considered by applying the center manifold theory. Numerical simulations reveal that the stability of Hopf bifurcation leads to the emergence of planktonic blooms. Moreover, it is found that the network structure can switch the types of spatiotemporal patterns, a new feature observed only in delay differential equations with network structure.